{
  "abstract": "On the $q$-quadratic lattices that carry the Askey--Wilson polynomials, the two roles played by a single Bernstein basis on the linear lattice, namely being a nonnegative partition of unity and carrying an orthogonal connection with the classical families, split between two distinct bases. The affine (spectral) basis retains the Bézier properties. A coprimality argument completely classifies the first-order lattice ladder: its solution space has dimensions two and one in degrees one and two, respectively, and is trivial in every degree at least three for every $q\\in(0,1)$. The affine factors do not satisfy the weight-shift mechanism that produces the $q$-Racah connection for the generalized-power basis. The generalized-power basis, built from products of two Askey--Wilson monomials, does not furnish a nonnegative partition of unity in a natural positive parameter region, specified in the text; there the unique normalisation summing to unity has sign-changing coefficients. Nevertheless, its connection coefficients with the Askey--Wilson polynomials are identified completely: they are $q$-Racah polynomials $R_m(μ(k);\\,ad/q,\\,bc/q,\\,q^{-n-1},\\,a/b\\,|\\,q)$ multiplied by fully explicit prefactors. The proof is bispectral, valid for every degree, and rests on the tridiagonal action of the Askey--Wilson operator on the generalized-power basis, with explicit band coefficients. A scaled limit recovers the Wilson/Racah connection, while a second limit gives the $q$-Hahn polynomial part of the normalised $q$-linear connection coefficients. A numerical case study on the NACA 2412 airfoil illustrates the orthogonal machinery.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04802",
  "author_total": 1,
  "authors": [
    {
      "name": "Iván Area"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7s8g4cz0yq01s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04802",
  "fulltext_url": null,
  "id": "7s8g4cz0yq01s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04802",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7197006868",
  "pdf_url": "https://arxiv.org/pdf/2608.04802",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7s8g4cz0yq01s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.04802",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04802"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bernstein-type bases on $q$-quadratic lattices and Askey--Wilson\\slash $q$-Racah connection coefficients",
  "title_text": "Bernstein-type bases on q-quadratic lattices and Askey--Wilson\\slash q-Racah connection coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7s8g4cz0yq01s",
  "venue": "arXiv",
  "versions": []
}